Lesson video

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Hello, I'm Mr. Langton.

And today we're going to create some area models, and use them to multiply fractions together.

All you're going to need something to write with, and something to write on.

Try and make sure you're in a quiet space with no distractions.

When you're ready we'll begin.

So we'll start with the Try this activity.

Yasmin is looking to lay tiles in her pantry.

The area of the floor is one metre squared.

She considers three different tiles.

How many of each tile would she need? I'll give you a minute to have a go, just pause the video when you're ready.

Give it a go, see what you can do, when you're ready unpause it and we'll do it together.

Pause in three, two, one.

Okay, how did you do? Let's start looking at it together.

Let's start with tile A.

Tile A is 25 centimetres across and 20 centimetres up.

So how many times could we fit that tile going across? If we're going across 25 centimetres, 25, 50, 75, 100.

We would fit four of those tiles going across.

And up to one metre it would be 20, 40, 60, 80, 100.

That would be five of them would go up.

So four across by five up, we would get 20 of those tiles in.

Now let's look at tile B.

Let's have a different colour, I'll do this one in blue.

Tile B.

So that's 50 centimetres across.

If we measure 50 centimetres across, we would fit two of those tiles going across.

And 10 centimetres up.

How many times does 10 centimetres go into a metre? Should go in 10 times.

So we're looking at 20 pink ones, we can also have 20 green tiles.

Now I know what you're thinking, you're thinking, well, if every answer has been 20 so far, this one's going to be 20 as well, isn't it? Always helps to check.

We'll start off let's go along the bottom.

We've already said if we split it into 10 centimetre gaps, then we're going to need 10 of them.

Now five centimetre going up.

Actually, how many times does five centimetres go into a metre? Well, if it takes two of them for every 10, so, 2, 4, 6, 8, 10, 12, 14, 16, 18, it's going to take 20 times.

So 10 lots of 20, is going to be 200 tiles, for tile C.

Two students are looking at this model to multiply 1/4 and 1/5.

The model's in the middle here in green.

I'm going to put a nice big square around it there, so you can see it.

So we'll be splitting into quarters going along.

Can you see that? Splitting it into fifths going down.

Now Carla, on the left, says that the blue rectangle is 1/4, which it is, and so the green block in the circle in the bottom left is 1/5 of 1/4.

We should also mention it's 1/5 multiplied by 1/4.

Now, Zaki says that the red rectangle, this vertical one on the left hand side, is worth 1/5.

So the green one, the one that's circled, is 1/4 of 1/5, or 1/4 multiplied by 1/5.

Now in both cases, this one in the corner that we've circled, which is one there out of 20 entire blocks.

So whichever calculation we do, we get 1/20.

What calculations do you think each of these show? Going to ask you to pause the video and have a quick go.

Make some notes and when you're done unpause it and we'll go through it together.

Pause in three, two, one.

Have you had a go? Let's look at this first one.

It's on the left hand side.

If we look going down it's been split into five equal parts, and we've shaded one of them.

So we've shaded 1/5 going down.

Going across it's been split into four equal parts, and we've shaded three of them this time.

So that's 3/4.

You could have written 3/4 multiplied by 1/5.

And you could have written of instead of multiply as well.

Now the crossover part is where we're going to get our answer.

There are three of them, and there are 20 altogether.

So 1/5 times 3/4, or 3/4 times 1/5, is 3/20.

Looking at the middle one, we're going down first.

Again, it's just split into fifths, but this time we've got four of those fifths.

Going across or in quarters, but it's only that bottom row that it's been shaded, so we've only got one of those quarters that has been shaded.

You could have got it the other way around.

You could have said that we've got 1/4 times 4/5.

Either way, the squares that have been shaded both times, there are four of them out of 20.

So 4/5 multiplied by 1/4, or 1/4 multiplied by 4/5, is 4/20.

If you're feeling really clever that wants simplifying.

How many times does four go into 20? It goes in five times.

That is 1/5.

1/4 of 4/5 is also 1/5.

Now for the last one.

If we look going down, it's split into five equal parts, we've shaded three of them.

Going across we've split it into quarters, and again we've shaded three of them, so 3/4.

Spirit pen wasn't working then.

And altogether that's 9/20.

Alternatively, you can say that we shaded 3/4, multiplied by 3/5, which once again is 9/20.

Then you start looking for patterns now of the multiplying.

So if we're multiplying two fractions together to get the answer can you see any connections? So if you look back at that first one.

And we said that we've got 1/5 times 3/4.

We multiplying numerators together, one times three is three, five times four is 20.

And if you check that's going to work every single time.

So we're going to use that for the next task.

What I'm asking you to do is pause the video to access the worksheets.

When you're ready unpause it and we'll go through it together.

Good luck.

How did you get on? Let's go through the answers now.

We'll start off with 1/2 times 2/3.

So we going to half the 2/3, or 1/2 times 2/3, we're splitting it into halves and thirds, which means that we're splitting it into six equal pieces altogether.

And we're going to be shading in two of those six.

So 1/2 multiplied by 2/3 is 2/6.

Which is also equivalent to 1/3 if you simplify it.

2/3 of 4/5, or 2/3 multiplied by 4/5.

If we're doing thirds and fifths multiplied together, we're splitting the box into 15 equal pieces altogether.

And if we've got 2/3 multiplied by 4/5, we're going to need eight of those.

Shall we shade that to make sure? If I shade in my 2/3.

And then in a different colour I'll shade in my 4/5.

Up to there.

We can see that these eight cross over part, two times four is eight, two multiplied by four is eight, so you've got 8/15.

And that one wants simplifying.

We've still got the diagram, but I haven't labelled it.

1/3 of 3/4, or 1/3 multiplied by 3/4.

So 1/3, can be only one of these three, isn't it? Still going across.

That there is 1/3.

I'll shade that in purple.

Moving the other way we want 3/4.

Let's have blue.

So that would be 1/4, 2/4, 3/4 there.

That's 3/4.

And as you can see the whole thing is split into 12 equal pieces, three multiplied by four is 12.

And we've got this crossover here where there are three of them.

One multiplied by three is three.

Now if you're feeling really clever, that's actually 1/4 of the shape and that cancels down to 1/4.

So the second part now, we don't have the models to help us, you might have drawn some.

If you didn't do let's just go through a few now.

1/2 multiplied by 1/5 is going to be 1/10.

1/5 multiplied by 1/2 is also 1/10, they're the same question, aren't they? 3/5 multiplied by 2/3, you may have got 6/15.

You may have been feeling quite clever and simplified it to 2/5.

3/7 multiplied by 9/10 is going to be 27/70.

By this point, we're getting beyond the stage of drawing diagrams now, aren't we? Because there's no way I'm going to draw a diagram that's got 70 boxes in it.

3/3 multiplied by 2/5.

Well, three multiplied by two is six, and three multiplied by five is 15, so it's 6/15.

Let's simplify that.

They both go into three, don't they? So that's 2/5.

Now, if you were really on the ball, you may have spotted that 3/3 is the same as a whole one.

So if we're doing one lot of 2/5, then of course the answer is 2/5.

Finally, 3/4 squared.

That means that if we square something we times it by itself, don't we? So 3/4 multiplied by 3/4 which is going to be 9/16.

How did you do? So we'll finish up with the Explore activity.

Up on the screen you've got two different diagrams, that end up with an answer of 12/143.

One of them is 4/11 multiplied by 3/13.

Another one is 2/11 multiplied by 6/13.

How many other pairs of fractions can you find that have a product of 12/143? Can you tell me what's the same about them, or what's different about them? Pause the video and have a go, and when you're ready unpause it, we'll go through it together.

Good luck.

How did you get on? I started by looking at the different ways that we can make 12 and the different ways we can make 143.

Now, I started with 143 because actually, the only way you can make it is 11 multiplied by 13.

So that's going to get me my denominator straight away.

It might seem a little bit cheeky, but that's how I've started off and I've set it up.

I'm trying to make 143 all the time, I must do 11 multiplied by 13.

Now I need to find as many different ways as possible to make 12.

I could do one multiplied by 12, that's going to work.

I could do two multiplied by six, and that one's already given to me, wasn't it? I could do three multiplied by four, and I could do four multiplied by three.

Now that one was given that was over there.

But it's important to bear in mind, that although I've still done three times four to get 12, or four times three to get 12, I've done 3/11 times 4/13, and I've done 4/11 times 3/13.

So they're very, very different questions.

Finally, I could have done 6/11 times 2/13, and that gives me 12/143.

And I could have an improper fraction, and do 12/11 times 1/13, and you get 12/143.

Then each of these could have been reversed as well.

So you could have 12/13 times 1/11, and so on.

So what's the same and what's different? Well, the same is that my denominators have got to be 11 and 13 every time.

Different is that the top ones could be different numbers.

One thing they've all got in common is that they must be pairs of factors of 12.